Yolanda Paid For Her Movie Ticket Using 28 Coins, All Nickels And Quarters. The Ticket Cost \$4. Which System Of Linear Equations Can Be Used To Find The Number Of Nickels, \[$n\$\], And The Number Of Quarters, \[$q\$\], Yolanda Used?A.

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Introduction

In this article, we will explore how to use systems of linear equations to solve a coin problem. We will consider the scenario where Yolanda paid for her movie ticket using 28 coins, all nickels and quarters, and the ticket cost $4. Our goal is to find the number of nickels, denoted as nn, and the number of quarters, denoted as qq, that Yolanda used.

Understanding the Problem

Let's break down the problem and understand what we are trying to solve. We know that Yolanda used 28 coins in total, and these coins are a combination of nickels and quarters. We also know that the total cost of the ticket is $4.

Defining the Variables

To solve this problem, we need to define two variables: nn for the number of nickels and qq for the number of quarters. We also need to define the cost of each coin. A nickel costs $0.05, and a quarter costs $0.25.

Setting Up the Equations

We can set up two equations based on the given information. The first equation represents the total number of coins used:

n+q=28n + q = 28

The second equation represents the total cost of the coins used:

0.05n+0.25q=40.05n + 0.25q = 4

Solving the System of Linear Equations

To solve this system of linear equations, we can use the method of substitution or elimination. Let's use the elimination method to solve for nn and qq.

Step 1: Multiply the First Equation by 0.05

We will multiply the first equation by 0.05 to make the coefficients of nn in both equations the same:

0.05n+0.05q=1.40.05n + 0.05q = 1.4

Step 2: Subtract the Second Equation from the First Equation

We will subtract the second equation from the first equation to eliminate the variable nn:

(0.05n+0.05q)−(0.05n+0.25q)=1.4−4(0.05n + 0.05q) - (0.05n + 0.25q) = 1.4 - 4

Simplifying the equation, we get:

−0.2q=−2.6-0.2q = -2.6

Step 3: Solve for q

We will solve for qq by dividing both sides of the equation by -0.2:

q=−2.6−0.2q = \frac{-2.6}{-0.2}

q=13q = 13

Step 4: Solve for n

Now that we have the value of qq, we can substitute it into the first equation to solve for nn:

n+13=28n + 13 = 28

n=28−13n = 28 - 13

n=15n = 15

Conclusion

In this article, we used a system of linear equations to solve a coin problem. We defined the variables nn and qq to represent the number of nickels and quarters used, respectively. We set up two equations based on the given information and used the elimination method to solve for nn and qq. The final answer is that Yolanda used 15 nickels and 13 quarters to pay for her movie ticket.

Final Answer

The final answer is:

  • Number of nickels: 15
  • Number of quarters: 13
    Frequently Asked Questions (FAQs) about Solving Coin Problems with Systems of Linear Equations =============================================================================================

Introduction

In our previous article, we explored how to use systems of linear equations to solve a coin problem. We defined the variables nn and qq to represent the number of nickels and quarters used, respectively, and set up two equations based on the given information. In this article, we will answer some frequently asked questions (FAQs) about solving coin problems with systems of linear equations.

Q: What is a system of linear equations?

A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables.

Q: How do I know which method to use to solve a system of linear equations?

There are several methods to solve a system of linear equations, including substitution, elimination, and graphing. The method you choose will depend on the specific problem and the variables involved.

Q: What is the difference between substitution and elimination methods?

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one variable.

Q: How do I know which variable to eliminate first?

When using the elimination method, you should eliminate the variable that has the smallest coefficient. This will make it easier to solve for the other variable.

Q: What if I have a system of linear equations with three or more variables?

If you have a system of linear equations with three or more variables, you can use the same methods as before, but you may need to use more complex techniques, such as matrix operations or Gaussian elimination.

Q: Can I use a system of linear equations to solve a problem with more than two variables?

Yes, you can use a system of linear equations to solve a problem with more than two variables. However, you will need to use more complex techniques, such as matrix operations or Gaussian elimination.

Q: How do I know if a system of linear equations has a unique solution?

A system of linear equations has a unique solution if the equations are consistent and the variables are independent. If the equations are inconsistent or the variables are dependent, the system may have no solution or an infinite number of solutions.

Q: What if I have a system of linear equations with no solution or an infinite number of solutions?

If you have a system of linear equations with no solution or an infinite number of solutions, you will need to re-examine the problem and the equations to determine the cause of the issue.

Q: Can I use a system of linear equations to solve a problem with decimal coefficients?

Yes, you can use a system of linear equations to solve a problem with decimal coefficients. However, you will need to use more complex techniques, such as matrix operations or Gaussian elimination.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about solving coin problems with systems of linear equations. We discussed the different methods for solving systems of linear equations, including substitution and elimination, and provided tips for choosing the right method. We also discussed how to handle systems of linear equations with more than two variables and how to determine if a system has a unique solution.

Final Answer

The final answer is:

  • Substitution method: Solve one equation for one variable and substitute that expression into the other equation.
  • Elimination method: Add or subtract the equations to eliminate one variable.
  • Matrix operations: Use matrices to solve systems of linear equations.
  • Gaussian elimination: Use a systematic method to solve systems of linear equations.